Date: Mon, 02 Dec 1996 14:42:19 GMT
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CSE 415  Prog. Assignment 3
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CSE 415      Programming Project #3     due May 17, 5pm
                                        (Use "Turnin" as directed).

        Here are the details I covered in class on Thursday, with a
more precise description of what you have to do. Some of the last
message on this is included here:



        Heuristic Selection of the "Best" Transformation in
        Proving Trigonometric Identities.

                                                (due Friday May 17)



        This project involves symbolic problem solving, creation of a
representation that facilitates implementation, and learning. The learning
part simply involves being able to use problems that have been solved by
the program to improve problem solving power later.
        The problem is to start with five simple trigonometric identities,
such as tan x = sin x/cos x, cot x = cos x/sin x, cosec x = 1/sin x,
sec x = 1/cos x, sin^2 x + cos^2 x = 1, and use them as operators to
transform the left side of a new trig equation, until the right side is
produced. The problem will be solved at that point.
        The first thing here is to find a way to represent the problem
and the operators (including operators which will be added as problems
are solved), so that programming will be as simple as possible. Then,
features of the left and right sides of the operators (and of the problem
identities) have to be chosen, which will help in both deciding which
operator to apply, and where to apply it.
        I will cover most of this in class, but you should think about it.
As an example, given the above five trig identities as operators, how
would you solve the problems:  Prove that tan x cot x = 1,
etc. Also, how do you write lisp functions to deal with all possible cases?
        To implement a problem-solver that would start with the five
basic identities, prove new identities, and then add those identities to
the set available for solving future problems, would require more
programming than you could do in a short time. So, you are going to
be given a set of initial identities (in standard form) and you will
have to write a lisp program to select the "best" transformation
identity for a given problem identity. You will make three, possibly
different selections, based on three different heuristics.
        Standard Form:

        Replace sin x with the symbol A, cos x with B, tan x with C,
        cot x with D, cosec x with E, and sec x with F.

        Remove all "-" signs by moving terms to the other side of
        the identity; remove all "/" signs by multiplying throughout
        by the denominator; remove all exponents by repeated
        multiplication; and remove all integer coefficients by
        repeated addition.
        For example, sin^2 x  +  cos^2 x  =  1  would become
        AA + BB = 1

        The final step is to lexically order each side of the identity
        and then to switch sides, if necessary, so that the left side
        is lexically before the right side.
        Thus, tan x  =  sin x/ cos x  would become  C  =  A/B, and then
                                                    A  =  BC

        Identities are restricted to trig functions of a single variable.
        The standard form is a sum of products form, with the product
        terms lexically ordered. This form greatly reduces the variability
        of representation for equivalent identities. It is a good example
        of a general rule in AI:  "Transform problem into a standard form
        so that the variety of equivalent forms is reduced as far as
        possible."

        Your program will assume that its inputs are trig identities
        given in the standard form.

        You will be given a list of identities which can be used as
        transforms, and one or more identities as problem identities.
        You should write three functions; one for each of the three
        heuristic methods defined below. Each function will then have
        to select the "best" transform identity to be first applied in
        solving the problem (ie in proving that the left side of the
        problem identity can be transformed into its right side). The
        three functions will usually pick different transform identities.

        The three heuristic methods will be named, "Simplification",
        "Familiarization" and the "Effectiveness Heuristic".

        Simplification Heuristic:

        Here the scoring function is the difference in the number of
        the symbols on the left and right sides of the transform
        identity. Before an identity can be selected, one of its sides
        must be applicable to the left side of the problem identity.

        eg  Problem:  Prove that sin^2x  +  tan^2x  +  cos^2x
                                 =  sec^2x

            Standard Form:   AA + BB + CC  =  FF

        Here the transform identity AA +  BB  =  1  can be applied
        to the left side of the problem, P, and the simplification
        score is 4 (there are 5 symbols on T's left side, including
        the + sign, and one symbol on T's right side). Applying T
        to the left side of P will reduce the number of symbols in
        P by 4.
                All of the T's with the same greatest score will be
        selected.
                Your program has to print out the scores for all of
        the T identities, for the given problem/s, and the
        identification numbers of the T identities.

        Familiarization Heuristic:

        The transform identities will be the first 16 identities
        in the list (Table I) given out in Packet #1. Their id
        numbers are also listed. You will use a list of these
        numbers to indicate the "familiarity score" of the
        transform identities. The first id no in the list show
        the most familiar identity, the second the next most
        familiar, etc.
        The transform identity chosen here will be the one closest
        to the head of the list, which can be applied to the left
        side of the problem identity.

        Effectiveness Heuristic:

        This one is more complex and is supposed to provide a score,
        called E, which is a measure of the applicability of either
        the left or right sides of the T identity to the left side of
        the P identity, and also the degree to which the T identity
        is likely to make the left side of the P identity more like
        the right side of the P identity. Thus it is an approximate
        measure of the degree to which T will make the left side of
        P (the current problem) become closer to the right side of
        P (the goal).

        Define the feature set of the left side of P to be:
        the set of single trig objects in Pl ( = P left-side),
        the set of all different pairs of trig objects in product
        terms of Pl, and the set of all pairs of product terms
        in Pl (where any common objects are taken away, "divided
        out"). Make a feature set for the left side of P, the
        right side of P, the left side of each T and the right side
        of each T. (The T feature sets will already be available).

        For each feature, we will use a weight to indicate its
        complexity. This will be w = sum of symbols in the feature.
        This will give added importance to matching features which
        have lots of symbols.

        eg  for the same P as above:  AA + BB + CC  =  FF

            Flp = {A,B,C,AA,BB,CC,AA+BB,AA+CC,BB+CC}
             w  = {1,1,1, 2, 2, 2,  5,    5,    5  }

             the feature set for the left side of P with the
             corresponding weight for each feature.

            Frp = {F,FF}
             w  = {1, 2}

        There will be available, similar feature sets for all of
        the available T identities.

        The E score is computed by adding together the weights of all
        features present in T with each weight multiplied by a
        "desirablity-factor", d, which is intended to show if the
        single feature associated with a given w is going to help
        or hinder finding a solution.

        So, to compute E, each feature in Flt  and in Frt is examined,
        and compared to the feature set of the left and right sides of P.
        Each such feature, fi will then have its weight, wi, and its
        desirability factor, di.

        Then E  =  Sum of wi*di  for all features in Flt  and Frt

        To find di, there are various possibilities, a feature could
        be in the left side of T and not in the right side of T, and
        it could be in the left side of P and not in the right side of P.
        This situation will be described by  1 0 1 0
        This is a very desirable situation, since applying the left side
        of T to the left side of P will remove a feature from the
        left side of P, and this feature is not in the right side of P,
        so this T will help solve the problem.

        The other situations, with their d scores, are:

        1 0 1 0  +4, 0 1 0 1  +4,  1 0 0 1  +4,  0 1 1 0  +4,

	1 0 0 0  -1,  0 1 0 0  -1,  1 1 0 0  -1,  1 1 0 1  +1

	1 1 1 0  +1,  1 1 1 1  +4

	(These d factors are quite arbitrary!)

	Using the same example as before:

	  	P:	AA + BB + CC  =  FF

		Flp = {A,B,C,AA,BB,CC,AA+BB,AA+CC,BB+CC}
		 w  = {1,1,1, 2, 2, 2,  5,    5,    5
	 	 
		Frp = {F,FF}
		 w  = {1, 2}

	Let's assume that the following T is being scored:

		T:	AA + BB  =  1

		Flt = {A,B,AA,BB,AA+BB}   Frt = {1}
	         w  = (1,1, 2, 2,  5  }         {1}


	The features from T will be scored one at a time, giving
	for A the situation is  1 0 1 0  here w = 1 and d = +4
	so the A feature adds +4 to E

	The features, B, AA, BB, AA+BB all have a 1 0 1 0 

	After they are considered, E = 4+4+8+8+20 = 44

	When the feature 1 is considered, it is 0 1 0 0 so d = -1
	and this contributes -1 to E
	so the effectiveness score, E, for applying this T to this P

	is  E  =  43


	Notes:

	For the familiarity heuristic, use the list:

	L  =  (4 1 2 10 6 5 3 14 11 8 9 15 12 13 7)

	The integers in the list correspond to the first 15 identities
	in the list from packet #1.

	Problem Identities to be used in your program:

	Find the best T identity , using each of the three above
	heuristics separately, for each of the following identities:-

	20,  28, and 38

	Your program should print out all of the scores computed and
	the id no of the T's selected. For each problem, only the 
	first T has to be found.
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